Dynamics of the fcc-to-bcc phase transition in single-crystalline PdCu alloy nanoparticles

Two most common crystal structures in metals and metal alloys are body-centered cubic (bcc) and face-centered cubic (fcc) structures. The phase transitions between these structures play an important role in the production of durable and functional metal alloys. Despite their technological significance, the details of such phase transitions are largely unknown because of the challenges associated with probing these processes. Here, we describe the nanoscopic details of an fcc-to-bcc phase transition in PdCu alloy nanoparticles (NPs) using in situ heating transmission electron microscopy. Our observations reveal that the bcc phase always nucleates from the edge of the fcc NP, and then propagates across the NP by forming a distinct few-atoms-wide coherent bcc–fcc interface. Notably, this interface acts as an intermediate precursor phase for the nucleation of a bcc phase. These insights into the fcc-to-bcc phase transition are important for understanding solid − solid phase transitions in general and can help to tailor the functional properties of metals and their alloys.

A solid-solid (s-s) phase transition is the transformation of a solid from one crystal structure into another 1 , and such a transition is common to many minerals 2 , metals 3 , alloys 3 , and ceramics [4][5][6] . Because the crystal structure of a solid determines its mechanical strength 7,8 , optical property 9 , and electrical 10 and thermal 11 conductivities, s-s transitions are among important processes in materials technology 12 . In metals and alloys, body-centered cubic (bcc) and face-centered cubic (fcc) structures are the two most common crystal structures. The fcc-to-bcc phase transition and vice versa are often used in the production of durable steels 12,13 , shape-memory alloys 14 , high entropy alloys 15 , and catalytic materials 16 . However, despite their technological significance, the details of these phase transition are largely unknown because exploring the nanoscopic intermediate stages of the transitions is both experimentally and theoretically challenging 17,18 .
Common experimental techniques to study phase transitions are X-ray diffraction (XRD) 19,20 , electron backscatter diffraction (EBSD) 21,22 , and conventional transmission electron microscopy (TEM) 15,23 . These experimental approaches have several limitations. For example, XRD can only assess the averaged crystallinity of materials and is blind to subtle intra-particle transformations. More importantly, in the case of martensitic transformations 24 , which is a critical process step in steel production (e.g., displacive fcc-to-bcc transformations), the phase transition occurs at ultrahigh speeds 12,25,26 . Thus, it is almost impossible to capture the dynamics of the transition with conventional EBSD and TEM approaches, which lack the necessary temporal resolution 15,21 . Furthermore, molecular dynamics simulations [27][28][29][30][31] , often used for the theoretical description of these transitions, provide very limited insights because the simulations employ oversimplified interatomic potentials 18 .
The crucial question for understanding fcc-to-bcc phase transitions is how one phase nucleates and grows within the other phase. To address this question, we examined how a single-crystalline PdCu alloy NP transforms from an fcc into a bcc NP using in situ heating TEM study. We chose this NP system for two reasons. First, the timescale for an fcc-to-bcc phase transition in PdCu alloy is sufficiently slow compared to steel 12,19 , which, in turn, allows us to image the dynamics of the transition with current state-of-the-art cameras. Second, the stress from the surfaces of single-crystalline NPs is significantly lower than the stress on individual grains of polycrystalline metals constrained by the surrounding grains 32 , enabling to probe the phase transition isolated from any external effects. A similar in situ TEM approach has been used to study many different phase transition phenomena in single-crystalline Pd 33 , AgPd 34 , Cu 2 S 35 , HfO 2 36 nanocrystals, and many other systems [37][38][39][40] .

Phase transition dynamics of an individual NP
We prepared PdCu alloy NPs with a metastable fcc phase using a onepot synthesis reaction, in which Pd 2+ and Cu 2+ precursor ions were reduced simultaneously to form the alloy NPs 16 . The NPs comprised 46% (at.) Pd and 54% (at.) Cu and had an average size of~8 nm (Supplementary Fig. 1). At this composition and below 505°C, the bcc phase is a thermodynamically stable phase ( Supplementary Fig. 2) 41 . Thus, upon sufficient heating, the as-synthesized metastable fcc NPs should transform into stable bcc NPs 16 . Figure 1 shows the structural transformation of an individual NP at 500°C. The spots in the fast Fourier transform (FFT) pattern of the initial NP (Fig. 1a: t = t 0 ) correspond to a single-crystalline fcc phase. In an fcc unit cell of PdCu alloy (inset in Fig. 1a: t = t 0 ), the Pd and Cu atoms randomly occupy the eight corners and six face-centered positions of the cell. After heating (to 500°C), the fcc NP evolves into a bcc NP as seen from its FFT spots ( Fig. 1a: t−t 0 = 400 s). Also, the presence of the forbidden 001 ð Þ bcc spot suggests that the alloy NP has an ordered bcc crystal structure 42 . Specifically, in this bcc unit cell designated as the B2 phase (inset in Fig. 1a: t−t 0 = 400 s), the Cu atom occupies the body-centered position while the Pd atoms are at the eight corners of the unit cell 41 . During the fcc-to-B2 transition ( Fig. 1a: t −t 0 = 238 s), the two different sets of spots in the FFT pattern correspond to coexisting fcc and B2 phases. Furthermore, the individual spots from two different phases are not distinctly separated; instead, they are connected with smeared traces (e.g., see traces between 020 ð Þ fcc − 110 À Á B2 and 002 ð Þ fcc − 002 ð Þ B2 spot pairs highlighted with white arrows in Fig. 1a: t−t 0 = 238 s). These traces between the spots of the two phases indicate that the fcc structure transforms into B2 structure gradually.
TEM and inverse FFT images of the NP in Fig. 1b, c show how the fcc phase evolves into the B2 phase. The B2 phase nucleates on the NP surface (Fig. 1b The few-atomic-layers-wide interface between the fcc and B2 phases is a coherent interface, meaning the lattice planes of the two phases are continuous across the interface ( Fig. 2c and Supplementary  Fig. 11c). This is notable because a coherent interface forms only when the two different phases have the same crystal structure with less than 10% lattice mismatch 43,44 . Earlier studies describing the fcc-to-bcc phase transition in steels, where the lattice mismatch between the two lattices is large, propose that these transformations should proceed via a highly mobile semicoherent interface and a set of dislocations 17,45 . Contrary to this general expectation, the fcc−B2 interface in PdCu system (with a large lattice mismatch of >20%) is coherent as it moves across the NP during the phase transition. The significance of the observed coherent interfaces is that the absence of defects between two phases is likely to be the reason as to why displacive transformations, common to steel 46 and shape-memory alloys 14 , proceed at such high speeds 12 .

Discussion of the phase transition pathway
The B2 phase propagates into the fcc phase by deforming the fcc crystal, as seen from the movement of the interface in Fig. 2c. This continuous lattice deformation during the phase propagation can also be verified from other view directions ( Supplementary Fig. 5). To obtain a three-dimensional perspective by unifying the observed structural changes viewed from different directions, we identified the fcc−B2 orientation relationship (see Supplementary Note 5 for more details) and summarized the phase transition pathway in Fig. 3. Here, a direct correspondence between the fcc and B2 crystal structures is established by constructing a body-centered tetragonal (bct) unit cell from two adjacent fcc unit cells ( Fig. 3a and Supplementary Fig. 6a), known as Bain correspondence 46 . During the phase transition, the constructed bct unit cell evolves into a B2 unit cell through a slight rotation and a change in the cell dimensions (from 2.7 × 2.7 × 3.8 Å 3 to  ð Þ bcc reflections (Fig. 1a). Here, Pd and Cu atoms undergo ordering into the B2 cell to occupy the corners and the body-centered position, respectively, via intra-cell atomic diffusion. Note that throughout the entire transformation, the cell volume does not change much (27.7 Å 3 vs. 27.0 Å 3 ; Fig. 3a).
The crystalline features of the fcc-to-B2 phase transition appear differently when viewed from different zone axes (Fig. 3b). When viewed from 010 ½ fcc and ½110 B2 zone axes (Fig. 3b: View 1), 002 ð Þ fcc planes change into 002 ð Þ B2 planes with the corresponding lattice spacings of 1.9 and 1.5 Å, and these lattice fringes rotate by~8°. Also, 200 À Á fcc planes become ð 110Þ B2 planes with the respective lattice spacings of 1.9 and 2.1 Å. These features are in line with our TEM observations shown in Fig. 1, 2. Projection from 10 1 Â Ã fcc and 1 1 1 zone axes (Fig. 3b: View 2) shows a 5°rotation between 020 ð Þ fcc and ð01 1Þ B2 planes, and a 10°rotation between 111 ð Þ fcc and ð 1 10Þ B2 planes (see Supplementary Note 2 for a more detailed discussion on how other view directions correlate with our TEM observations). It is important to note that Bain correspondence was proposed in 1924 and has been widely used ever since to describe the fcc−bcc phase transitions despite lacking direct experimental evidence for the transformation 45,46 . Our direct observations from different projected views reveal the details of Bain transformation.
To understand the fcc-to-B2 phase transition from a thermodynamic perspective, we evaluate the bulk and surface energies of fcc and B2 phases using density functional theory (DFT) calculations ( Fig. 4 and Supplementary Table 2). The energy of the bulk B2 crystal is 36 meV/atom lower than that of the bulk fcc crystal (i.e., E fcc!B2 = E fcc À E B2 = 36 meV/atom), suggesting that the B2 phase is the stable phase. This is in line with previous studies 47,48 and the Cu−Pd phase diagram (Supplementary Note 1) 41 . To explain our observed phase transitions, we note that E fcc!B2 can be interpreted as the thermodynamic force driving the metastable fcc phase into B2 phase (at 350−500°C). This driving force for the phase transition increases significantly in the presence of surfaces (E fhklg fcc !fijkg B2 = 64 À 304 meV/ atom) compared to the bulk case (E fcc!B2 = 36 meV/atom), as shown in Fig. 4a, where commonly observed facets between the two phases are compared. Note that the transformation of f220g fcc into f211g B2 surface has the highest driving force of 304 meV/atom. This is consistent with the TEM observation in Fig. 1, showing that the phase transition initiates and proceeds along the 022 ð Þ fcc and 112 À Á B2 crystal planes. Furthermore, in line with general expectations, the driving force for the phase transition is larger for smaller NPs (with larger surface-tovolume ratios; Fig. 4b). In more general terms, the larger driving force for the surfaces implies that the new B2 phase should nucleate from the surface and not from the core of a NP.

Nucleation of the B2 phase
In order to explore how an fcc-to-bcc phase transition initiates, we tracked the nucleation of a B2 phase within an fcc NP. During our heating studies, when the temperature was increased from 350 to 500°C at a rate of 1°C/s (Fig. 5a), the 002 ð Þ fcc planes of the original fcc crystal (Fig. 5b: t−t 0 = 12 s) start to bend at the top left corner of the NP, as shown in Fig. 5b. First, the 002 ð Þ fcc planes rotate by 2° (Fig. 5b: t−t 0 = 40 s), and then bend further until the angle reaches 5° (Fig. 5b:   Â Ã B2 zone axes (View 2). Red spheres represent either Pd or Cu atoms in a disordered fcc unit cell, while blue and orange spheres correspond to Pd and Cu atoms in the B2 unit cell, respectively. Fig. 2c and serves as a precursor phase which then transforms into a B2, and the growth direction of the B2 phase follows the propagation of the interface into the fcc NP. Note that the newly formed B2 phase clearly displays 101 À Á B2 and 110 À Á B2 lattice planes (Fig. 5a: t−t 0 = 167 s), and the transformation of the fcc into the B2 phase at the corner of the NP is consistent with the projected view expected from 10 1 Â Ã fcc and 1 1 1 Â Ã B2 zone axes (Fig. 3b: View 2). The fact that coherent interface forms prior to the nucleation of the B2 phase, serving as a prenucleation phase for B2, is in stark contrast to earlier studies proposing the nucleation of a bcc phase to occur either via twining within the fcc structure 49 , or the formation of stacking faults from fcc dislocations 50 .
As a final remark, it is important to mention that single-crystalline NPs used in our study constitute the simplest model system, which while suitable to capture the essence of the phase transition, may deviate from the transitions that occur in bulk polycrystalline materials in several ways. In bulk, the nucleation usually starts from the grain boundaries, dislocations, or defects instead of free surfaces in the NP case. Second, the phase transitions and volume changes of the individual grains are constrained by the surrounding grains; thus, strains and dislocations could arise to accommodate the volume changes. Nevertheless, because of ultrahigh speeds at which many displacive transformations take place, it is very likely that (dislocation-free) coherent interfaces are more common in these phase transitions. Furthermore, our study shows that such coherent interfaces not only play an important role in the propagation of a new phase during s-s phase transitions but are also critical for its nucleation as well.

Synthesis of fcc PdCu NPs
The following reagents were purchased from Sigma-Aldrich Co. The PdCu alloy NPs used in our study were synthesized based on a modified procedure of Tong et al. 16 . Specifically, 5.0 mg of Na 2 PdCl 4 , 6.6 mg of Cu(acac) 2 , 11.0 mg of ribose, 1.3 mL of ODE, and 1.3 mL of OAm were mixed in a 20 mL capped glass vial and ultrasonicated for 30 min to dissolve the solid chemicals (i.e., Na 2 PdCl 4 , Cu(acac) 2 and ribose) in the mixed solvent solution (i.e., ODE and OAm). Next, the vial containing the homogeneous solution was heated from room temperature to 180°C in an oil bath and kept at this temperature for 180 min during which the color of the final solution turned black. Then synthesized NPs were washed by centrifugation with hexane/ethanol mixture (at a volume ratio of 1:1) five times and then dispersed in hexane before the use.

In situ TEM experiments
Two TEMs were used for in situ heating studies: Thermofisher Titan S/ TEM (Thermo Fisher Scientific Ltd., Hillsboro, OR, USA) equipped with a Bruker Xflash 6 T | 30 energy-dispersive X-ray (EDX) spectrometer (Bruker, Billerica, MA, USA) and an aberration-corrected JEOL ARM300F (JEOL Ltd., Tokyo, Japan). Both TEMs were operated with an accelerating voltage of 300 kV. The Wildfire heating holders (DENSsolutions, Delft, Netherlands) were used for heating the NPs during in situ observations. Typical electron fluxes used for in situ imaging were in the range of 200−6000 e − Å −2 s −1 . Image series were acquired with two different cameras: a Gatan K2 IS camera (Gatan Inc., Pleasanton, CA, USA) on the Thermofisher Titan S/TEM and a Gatan One-View camera (Gatan Inc., Pleasanton, CA, USA) on the JEOL ARM300F. Inverse FFT images in Fig. 1c were obtained by separately filtering and then combining fcc and B2 spots in FFT patterns in Fig. 1a, and the same processing was used for Supplementary Figs. 3, 4, 10, and 14.

DFT computations
All plane-wave DFT computations were performed with the PWSCF program available in the Quantum Espresso suite 51,52 . Here, Perdew-Burke-Ernzerhof (PBE) functional under the generalized gradient approximation (GGA) framework was used to approximate the exchange-correlation energies 53 . Electron-ion interactions were modeled with the projector augmented-wave (PAW) method. Pseudopotentials associated with the PBE functional and the PAW method were obtained for Cu and Pd from the pslibrary 54 . The kinetic energy cutoffs for wavefunctions and charge densities were set to 50 and 500 Ry, respectively. Gaussian smearings, necessary for modeling metallic systems, were employed with a width of 0.01 Ry. The bulk B2 phase was modeled with the experimentally reported 3.0 Å × 3.0 Å × 3.0 Å unit cell with 8 × 8 × 8 Monkhorst-Pack (MP) sampling of the Brillouin zone 55 . To mimic a disordered fcc structure, we modeled it with a 32atom special quasi-random structure (SQS) supercell with 4 × 4 × 4 MP sampling 56,57 . The surface slabs for fcc and B2 structures were generated using the pymatgen package 58 with a minimum slab thickness of 5 Å and minimum vacuum thickness of 10 Å. The MP sampling of the : A f110g B2 : A f211g B2 = 1 : 1 : 1 and the average surface energy of E A = 227 meV/atom).

Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability
The data that support the findings of this study are available from the corresponding author upon request.  Supplementary Fig. 9a for the unprocessed TEM images). The fcc, B2, and interface regions are false-colored in red, blue, and orange, respectively. Note that the nucleation of the B2 phase starts with the formation of an interface region, which acts as a precursor phase for the nucleation. The dashed white lines are guides showing the gradual bending of 002 ð Þ fcc planes into 101 À Á B2 planes as the B2 phase nucleates and grows. The dashed red lines are the 002 ð Þ fcc planes. t 0 is the timepoint at which we started increasing the temperature from 350 to 500°C at a rate of 1°C s −1 .